a proof by PowerOfPi
We will prove that there are infinitely many Pythagorean triples where the Pythagorean triples are primitive (the three numbers are relatively prime) and the three numbers are integers.
First, we will mess around with :
Now, we will set some restrictions to help simplify. We set triple. Now this is what we have after plugging in for :
, like in the
So as long as is odd, then there is an integer value for , making an integer value for .
We know the only time a product is odd, is when both the numbers being multiplied are odd. Thus, if is odd, then must be odd as well. But can be any odd number? No. If
is one, then we have:
But we cannot have as a side of a triangle. So examples:
. Now we will try some more

We have arrived at some common triples. So every other odd number works, and there are infinitely many odd numbers, so there are infinitely many Pythagorean triples.
We can do a similar thing as above with and having a difference of , , , and so on.
1.
Find a Pythagorean triple where
2.
Find a Pythagorean triple where
3.
Find a Pythagorean triple where
.
.
.